Gambling, Risk Appetite and Asset Pricing

27 Pages·2017·0.37 MB·English

Gambling, Risk Appetite and Asset Pricing*
Carlos Carvalho Daniel Cordeiro Ruy Ribeiro
Central Bank of Brazil PUC-Rio PUC-Rio
PUC-Rio
Eduardo Zilberman
PUC-Rio
March 2017
Abstract
We show that time variation in gambling activity in the US provides relevant
information for asset pricing. Adding a conditioning variable based on gambling
activity improves the fit of both CAPM and consumption CAPM. In particular,
it performs better than using Lettau and Ludvigson (2001)’s consumption-wealth
ratio (cay) as conditioning variable. Moreover, this measure helps forecasting excess
returns at longer horizons, providing information that is not already contained in
either price-dividend ratio nor consumption-wealth ratio. Moreover, we find that
ourgamblingvariablehasbetterout-of-samplepredictiveperformancethanallother
alternative variables. [Preliminary, Do not circulate without prior consent]
*For comments and suggestions, we thank Marcelo Medeiros and Walter Novaes. All errors are ours.
The views expressed in this paper are those of the authors and do not necessarily reflect the position of
the Central Bank of Brazil. Carvalho: [email protected]. Cordeiro: [email protected]. Ribeiro
(corresponding author): [email protected]. Zilberman: [email protected].
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1 Introduction
Many asset pricing studies argue that the relation between expected returns and risk
changesovertimeimplyingthatmarketexpectreturnsarepredictableandthatthecross-
section of expected returns depends on the state of the economy. On the theoretical side,
changes in expected returns in the overall market and across assets could be justified
for example by a change in relative risk aversion (Campbell and Cochrane [1999] in the
context of a habit formation model), changes in market risk (for example, in the case
of the long-run risk model of Bansal and Yaron [2004]), or changes in other risks. In
equilibrium, price ratios such as the price-dividend ratio, and consumption-wealth ratio
would vary over time to capture these changes. Many papers have shown that price
ratios are useful in forecasting market excess returns (see Cochrane, 2001 for a textbook
treatment), but many econometric issues are also raised against these results (Welch and
Goyal [2008]).
It is however appealing to many financial economists that asset pricing relations could be
dependent on the state of the economy. For instance, Lettau and Ludvigson [2001b] use
a estimated consumption-wealth ratio in conditional CAPM and consumption models,
while Lettau and Ludvigson [2001a] also use it in predictive regressions. We take these
potential conditional variables to be proxies for a broad definition of risk appetite in the
economy. Both market risk premium and the dispersion of expected returns in the cross-
section would be wider in times of lower risk appetite. Practitioners and financial press
usually refer to the idea of risk appetite when discussing the pricing of financial assets.
The market’s appetite for risk would depend obviously on the reward-risk trade-off that
is present in the financial assets, but also on other risks that affect investors’ decision
making. It is obviously very hard to measure the willingness to take risk in the overall
economy over time without relying on asset prices. In theory, we would like to observe
directly the willingness to take risk and how it changes over time. An alternative would
be using an asset with a stable risk and return profile.
In this paper, we use a long time series of gambling in the US to construct a measure
of risk appetite that varies over time at a quarterly frequency. We show that adding
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a conditioning variable based on gambling activity improves the fit of both CAPM and
consumption CAPM. In particular, it performs better than using Lettau and Ludvigson
(2001)’sconsumption-wealthratio(cay)asconditioningvariable. Moreover,thismeasure
helps forecasting excess returns at longer horizons, providing information that is not
already contained in either price-dividend ratio nor consumption-wealth ratio. Moreover,
we find that our gambling variable has better out-of-sample predictive performance than
all other considered variables.
One of the advantages of casino data is that most of the gambling options have constant
risk and constant reward, making it a good “asset” in the estimation of risk appetite.
One disadvantage is that gambling has negative expected return. Therefore, we need to
assume that (some) investors that choose to go to casinos realize utility gains directly
fromgamblingactivity,makingthesubjectiveexpectedreturnpositive. Alternatively,we
could also consider alternative preferences that are not globally concave and that derive
utility from positive skewness in the spirit of Friedman and Savage [1948].
Our risk appetite measure is the cointegrating error of the relationship between casinos’
gambling revenue and other variables, such as airline fares, aggregate income and a
measures of the size of the gaming industry. The idea is to control for other variables
thataffectdemand andsupplyofgambling, butarearguablyunrelated to riskingeneral.
We focus on casino gambling rather than gambling in general for two reasons. First, the
reward-risktrade-offremainsconstantovertimeformostofthegames, suchasblackjack,
roulette,andsoon. Hence,asopposedtootherlotteriesingeneral,thechangeinrevenues
are not likely to be driven by changes in the reward-risk trade-off over time. Second, the
rewards in casino gambling do not depend on information, such as in the case of sports
betting and racing.
Ourriskappetitemeasureisnoisy,asthecointegratingerrormayencapsulateinformation
onotherdeterminantsofgamblingrevenue. Hence,itisreassuringthatthisnoisymeasure
improves the fit of classical asset pricing model and also helps predicting excess market
returns at longer horizons. Hopefully, a cleaner measure would perform even better in
describing the dispersion and the evolution of expected returns. Bias.
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Empirically, it is hard to measure aggregate risk appetite or risk aversion precisely over
time. The existing approaches involve the use of surveys or hypothetical lotteries in
particular points in time, or financial assets in an attempt to back out such measures.
The former approach does not yield long time series of risk appetite and does not provide
information on the aggregate willingness to take risks, whereas measures based on asset
prices are contaminated as prices are not solely determined by risk aversion and risk, but
also by cash flow expectations that vary over time too. In that sense, our risk appetite
measure circumvent these problems with a long time series that is independent of asset
prices, that conveys aggregate information on risk and risk aversion and that has stable
payoffs.
The remainder of the paper is organized as follow. First, we define our conditional
variable and show how we estimate it. Second, we show that this conditional variable
does provide useful predictive information both in-sample and out-of-sample. Third, we
present evidence that our risk appetite measure is relevant in explaining the cross-section
of asset returns.
2 Risk Appetite and Gambling
Wedefineriskappetitebroadlyasastatevariablethatgovernsthetrade-offbetweenrisk
and return. If risk appetite is high (low), investors are willing to take more (less) risk for
a given expected return. Hence, this measure should include all variables that somehow
affect this trade-off, such as the degree of risk aversion (i.e. market-wide willingness
to take risk), background risk (i.e. volatility of non-tradeable income), asset risk (i.e.
volatility of market and portfolio returns), and so on.
Our crucial assumption is that risk appetite not only governs the trade-off between risk
and return in the financial market, but also affects the decision to take non-market risk
such as casino gambling. In this section, we describe how we use a long time-series of
casino gambling activity in the US to construct a proxy of risk appetite that varies over
time at a quarterly frequency. In the next sections, we show that this measure improves
thefitofbothCAPMandconsumptionCAPM,aswellashelpsforecastingexcessreturns
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1.5
1
0.5
s
nit
U 0
d
e
z
di
r
a −0.5
d
n
a
St
−1
−1.5
−2
55 60 65 70 75 80 85 90 95 00 05 10 15 20
Year
Figure 1: Evolution of CASINO from 1959Q1 to 2016Q4.
at longer horizons.
Figure 1 plots our measure of casino gambling activity, which is the log of real casino
gambling expenditures per capita in the US (CASINO) over time demeaned and nor-
malized. Data on casino gambling expenditures were obtained from the NIPA tables
(code DCASRC0). We should note that casino activity increased steadily during the
1960s and 1970s. It decelerated during the 1980s and, then, accelerated again during the
1990s. Finally, after 2000, CASINO seems to have either increased at a very small pace
or stabilized. Before 1981, the series is very volatile relative to the later period.
The advantages of using this time series to extract a proxy for risk appetite are the fol-
lowing. First, as opposed to other lotteries in general, most casino gambling options (e.g.
blackjack or roulette) have roughly stable risk and reward over time. Hence, although
gambling technologies as well as the distribution of the games might have changed over
time, short-to-medium run movements in demand for casino gambling relative to income,
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rather than other financial assets, are likely to reflect changes in risk appetite. In addi-
tion, the rewards in casino gambling do not depend on information, such as in the case
of sports betting and racing. Second, CASINO is a long time-series that encompasses
the whole US territory, allowing us to extract a long time-series that proxies for risk ap-
petite in the US. Finally, as opposed to many other conditional variables used in CAPM
and consumption CAPM or to forecast returns, such as cay or price-dividend ratio, our
proxyforriskappetitedoesnotdependonassetprices, whichmaybeasourceofspurious
results.
Of course, there are also disadvantages. First, gambling has negative expected return
and, thus, a standard expected utility investor would not gamble. Hence, we implicitly
assume that investors that choose to go to casinos realize utility gains directly from
gambling activity, making the subjective expected return positive or that utility is not
globally concave. Second, the skewness of gambling returns may differ from that of asset
returns. Third, the distribution of characteristics (age, sex, social class, etc.) of casino
gamblers might change over time. It might, as well, differ from that of investors. Finally,
despite our effort to filter for other non-risk determinants of casino expenditures, our
risk appetite measure is noisy as it encapsulates information on other determinants of
gambling revenue. All these concerns may bias our measure of risk appetite. Despite this
potential bias, it is reassuring that our proxy for risk appetite improves the fit of both
CAPM and consumption CAPM, as well as helps forecasting excess returns at longer
horizons.
OurriskappetitemeasureisthecointegratingerroroftherelationshipbetweenCASINO,
aggregate income measured by the log of real GDP per capita (GDP), the log price index
associated with air transportation (deflated by the CPI) obtained from the BEA (AIR-
FARES),1 and the number of states in which casino gambling are legalized (STATES).2
The idea is to expunge long-term variations related to the size of the of casino gambling
1We also obtained an annual inflation adjusted measure of the average of domestic round-trip fares
in the US at the “Airlines for America” website (http://airlines.org/dataset/annual-round-trip-fares-
and-fees-domestic/), constructed based on the the Airline Origin and Destination Survey (DB1B). We
interpolated lineraly this series to make it quarterly. Our results are robust to the use of this series
instead.
2In the appendix, we describe how this variable was constructed.
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2
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s
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d 0
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di
ar −0.5
d
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a
St −1
−1.5
CASINO
GDP
−2
STATES
AIRFARES
−2.5
80 82 85 87 90 92 95 97 00 02 05 07 10 12 15 17
Year
Figure 2: Evolution of CASINO, GDP, AIRFARES and STATES from 1981Q1 to 2016Q4.
market that goes beyond GDP, but are arguably unrelated to risk in general. In partic-
ular, the steadily decline of airfares after the deregulation of the airline industry in 1978,
as well as the increasing number of states where casino gambling is a legal activity, are
proxies that capture the trend of the size of the casino gambling market over time.
As mentioned above, CASINO is very volatile before 1981 and its trend is significantly
different. Hence, we restrict the sample estimation from 1981Q1 to 2016Q4. Figure 2
plotstheevolutionofCASINO,GDP,AIRFARESandSTATESfrom1981Q1to2016Q4.
We consider several specifications to estimate the cointegrating system described above.
We estimate it by ordinary least squares (OLS), dynamic OLS (DLS) with lags and leads
varying from zero to eight. Results are reported in Table 1. Note that the coefficient
associated with GDP varies between 1.2 and 1.4, depending on the model used. This
is expected as casino is arguably a luxury good. Hence, one might expect that the
“elasticity” between gambling activity and GDP is above one. In addition, the coeffi-
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Table 1: Cointegrating Equations
Model lags / leads CONSTANT GDP AIRFARES STATES
OLS - -9.167 1.403 -0.311 0.074
DLS 0 -8.675 1.358 -0.317 0.075
DLS 1 -8.350 1.345 -0.354 0.075
DLS 2 -8.310 1.351 -0.376 0.075
DLS 3 -8.119 1.354 -0.422 0.074
DLS 4 -7.993 1.363 -0.468 0.073
DLS 5 -7.373 1.320 -0.506 0.074
DLS 6 -6.864 1.298 -0.566 0.074
DLS 7 -6.640 1.282 -0.579 0.075
DLS 8 -7.084 1.293 -0.514 0.077
Nota:Aprimeiralinhaapresentaoscoeficientesdaregressa˜oOLSeaslinhasseguintes
apresentam os par^ametros obtidos por DOLS (”Dynamic OLS”), seguindo Stock and
Watson [1993], com lags e leads das varia´veis independentes. Na˜o foram colocados
”lags”e”leads”davaria´velLC,porqueessavari´aveltempoucavaria¸ca˜o. Aprimeira
diferenc¸adessevetor´eumasequ^enciade216zerose16uns,assimseforemadiciona-
dos v´arios lags e leads dessa varia´vel, haver´a um problema de multicolinearidade
cient associated with AIRFARES is negative as expected, varying between -0.3 and -0.6.
Finally, the coefficient associated with STATES is positive (approximately 0.075) as ex-
pected and remarkably robust across specifications. In addition, Table 2 reports several
Johansen cointegration tests rejecting the null of no cointegration. These tests indicate
the existence of one cointegration restriction.
In what follows, we consider the specification DLS with four leads and lags to account for
any impact of misspecified seasonal adjustments in the data. In particular, we discuss to
whatextentourmeasureofriskaptetiteimprovesthefitofbothCAPMandconsumption
CAPM models and helps forecasting excess returns. Importantly, in the appendix, we
report results considering the other specifications, which are remarkably robust.
3 Predictability and Risk Appetite
In this section we evaluate how our risk appetite measure (rap) helps predict excess re-
turns, comparing its performance with other standard measures used in the literature,
namely, price-dividend ratio (d/p) and consumption-wealth ratio (cay). Since, as ex-
plained above, we restrict rap to start at 1981Q1, we consider the sample from 1981Q1
to 2015Q4 and report both in-sample (IS) and out-of-sample (OOS) prediction statistics
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Table 2: Johansen Cointegration Test
Max Eigenvalue Trace H0=r
Test Statistic 95%CL Test Statistic 95%CL
PANEL A - lag interval: 1
46.70 28.59 81.10 54.08 0
16.06 22.30 34.41 35.19 1
14.87 15.89 18.35 20.26 2
3.47 9.16 3.47 9.16 3
PANEL B - lag interval: lags 1 a 2
35.81 28.59 66.92 54.08 0
14.73 22.30 31.10 35.19 1
11.24 15.89 16.37 20.26 2
5.13 9.16 5.13 9.16 3
PANEL C - lag interval: 1 a 3
38.46 28.59 70.51 54.08 0
18.81 22.30 32.05 35.19 1
9.71 15.89 13.24 20.26 2
3.53 9.16 3.53 9.16 3
PANEL D - lag interval: 1 a 4
35.09 28.59 66.67 54.08 0
17.56 22.30 31.58 35.19 1
10.19 15.89 14.01 20.26 2
3.82 9.16 3.82 91.65 3
Nota: Osresultadosapresentadosnessatabelana˜oassumemtend^encialinear
nosdados, masassumequearela¸ca˜odecointegra¸ca˜otemumaconstanteJo-
hansen [1988]. O teste de Johansen testa a hipo´tese H0 de haver r rela¸co˜es
de cointegra¸ca˜o contra a hip´otese de haver H1 = r+1 rela¸co˜es de cointe-
grac¸ao. A primeira coluna estat´ıstica teste informa o valor da estat´ıtica de
teste para o teste do autovalor ma´ximo e a segunda informa o valor da es-
tat´ısticadetesteparaotestedotra¸co. 95%VCinformaovalordaestat´ıstica
teste para o intervalo de confian¸ca de 95%. Ambos os tetes apresentam forte
evid^encias de haver uma u´nica rela¸ca˜o de cointegra¸ca˜o entre 𝑔𝑎𝑐 , 𝐿𝐶 , 𝑝𝑖𝑏
𝑡 𝑡 𝑡
e 𝑐𝑣 intedependente da especifica¸ca˜o do nu´mero de lags.
𝑡
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concerning the period from 1991Q1 to 2015Q4.
Table 3 below reports some statistical properties of the aforementioned measures. Notice
that cay and d/p present a stronger correlation than that of cay (or d/p) and rap. In
addition, although high, the persistence of both rap and cay is fairly smaller than that of
d/p. Hence, theinferenceproblemsthatariseinforecastingequationswhenthepredictor
is very persistent is mitigated once rap or cay is considered.
Table 3: Correlation and statistics
Panel A: Correlation matrix
cay d/p rap
cay 1
d/p 0.31 1
rap -0.10 0.05 1
Paneal B: Statistics of variables
mean 0.0055 -3.7534 -0.0008
Standard deviation 0.02 0.39 0.08
Autocorrelation 0.90 0.97 0.89
Nota: Estat´ısticas das vari´aveis utilizadas para previsa˜o.
Essas estat´ısticas s˜ao referentes ao per´ıodo de 1981T1-
2015T4.
Inordertocompareperformanceacrossmeasures, werunthefollowingsetofregressions:
𝑒𝑥𝑐𝑒𝑠𝑠 = 𝛼+𝛽𝑋 +𝜀 ,
𝑡+ℎ 𝑡−1 𝑡
where 𝑒𝑥𝑐𝑒𝑠𝑠 measures excess returns accumulated over ℎ quarters associated with
𝑡+ℎ
the value-weighted CRSP Index. The risk-free rate is the return associated with the
three-month U.S. Treasury bill. Moreover, 𝑋 is one of aforementioned predictors or a
𝑡
vector of some of these variables. Notice that one advantage of our measure is that,
as opposed to both d/p and cay, we do not use prices to compute it. The use of asset
prices to compute both independent and dependent variables might hinder the economic
interpretation behind the estimated relation between them.
The next subsections compare the predictive capacity in-sample (IS) and out-of-sample
(OOS) between rap, d/p and cay.
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